We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical ideas applied to study the oscillatory behavior of the error term in the prime number theorem.

中文翻译：

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非实数极点和分布不规则

我们研究加权狄利克雷级数的一般理论及其系数的相关求和函数。我们表明，任何非实数极点都会导致振荡误差项。即使存在无限多个具有相同实部的非实极点，这也适用。此外，我们考虑非实极位于一条直线附近但不在一条直线上的情况。证明方法是经典思想的推广，用于研究素数定理中误差项的振荡行为。